(CPG Bagels) CPG Bagels starts the day with a large production run of bagels. Throughout the morning, additional bagels are produced as needed. The last bake is completed at 3 pm and the store closes at 8 pm. It costs approximately $0.20 in materials and labour to make a bagel. The price of a fresh bagel is $0.60. Bagels not sold by the end of the day are sold the next day as 'day old' bagels in bags of 6, for $0.99 a bag. About two-thirds of the day-old bagels are sold; the remainder are just thrown away. There are many bagel flavours, but for simplicity, concentrate just on the plain bagels. The store manager predicts that demand for plain bagels from 3 pm. Until closing is normally distributed with mean of 54 and standard deviation of 21.

a. How many bagels should the store have at 3 pm to maximize the store's expected profit (from sales between 3 pm until closing)? (Hint: assume day-old bagels are sold for $0.99/6= $0.165 each. don't worry about the fact that day-old bagels are sold in bags of six.)

b. Suppose that the store manager is concerned that stock outs might cause a loss of future business. To explore this idea, the store manager feels that is is appropriate to assign a stock out cost of $5 per bagel that is demanded but not filled. (Customers frequently purchase more than one bagel at a time. This cost is per bagel demanded that is not satisfied rather than per customer that does not receive a complete order.) Given the additional stock out cost, how many bagels should the store have at 3 pm to maximize the store's profit?

c. Suppose the store manager has 101 bagels at 3 pm. How many bagels should the store manager expect to have at the end of the day?